Film the bottom of a 6-quart pot with olive oil and heat over medium-high. Stir in the onions, season with salt and pepper and cook until the onions are golden, 5 to 8 minutes, stirring often. Add the garlic and basil and cook briefly, until garlic is fragrant.

Add the tomatoes and broth and bring to a simmer. Cover and cook 5 to 10 minutes. Taste and check the seasoning. Cool 10 minutes, then lightly puree in a food processor or blender or in the pot using an immersion blender. Serve hot.